The Logical Study of Argument: Propositional Logic
Logicians aim to develop simple and easy-to-use methods to distinguish good arguments from
bad arguments. When you study deductive logic, good arguments and bad arguments differ from
one another in how their premises relate to their conclusion. It is of the utmost importance to
appreciate that the logical study of argument does not care whether the premises of an argument
are true or false. A good deductive argument is one whose premises, if true, guarantee the truth
of its conclusion; it does not matter whether the argument’s premises are in fact true. Likewise,
a bad deductive argument is one whose premises, even if true, offer no guarantee that the
conclusion is true. These are called valid and invalid arguments, respectively.
We can provide more precise definitions of these two fundamental terms as follows:
Good Argument Valid =df
(a) it is impossible for the premises to be true & the conclusion false
(b) if the premises are true, then conclusion must be true as well
Bad Argument
Invalid =df
(a) it is possible for the premises to be true & the conclusion false
(b) even if the premises are true, the conclusion might be false
Chart 1: The Definitions of Valid and Invalid Arguments
To study validity, logicians turn to formal languages which provide them with the tools to cut
away the extraneous extra stuff we find in natural language arguments like English and reveal
the bare bones, underlying logical form of an argument. For, it is this – an argument’s logical form
– that determines whether it is valid or invalid, as defined above in Chart 1.
This document offers you a primer on the formal language that will take up the bulk of our study
in this class – viz., propositional logic. Once you have the fundamentals of this formal language
in hand, you will then learn how to use this formal language to (1) represent the underlying logical
form of natural language arguments and (2) perform simple, rule-governed tasks to determine
whether it is valid or invalid.
I shall refer to (1) as the process of translation and (2) will take a few different forms as we move
into later units of the class. At present in Unit 1, we are going to learn the method of truth tables
or semantic tableaux. However, throughout Units 2, 3 and 4, we will replace this truth table
method with a system of deduction rules to construct proofs in propositional logic. These two
methods will accomplish (2) above.
Part 1: The Symbolism of Propositional Logic
(1) Propositional Variables (A, B, C, … X, Y, Z)
Upper-case letters stand for (a) simple, (b) declarative, (c) positive sentences of English. In
particular, what is important to appreciate is that the propositional variables A, B, C, … X, Y, Z are
to be used in the place of complete sentences. Sentences have truth-values and there are exactly
two such values – true and false – and we assume that every sentence takes one (and only one)
of these two values. Here are a few examples to help convey this point.
Example1
Moby Dick is a whale
“W” stands for [Moby Dick is a whale]
Example2
Captain Ahab hates Moby Dick
“H” stands for [Captain Ahab hates Moby Dick]
Chart 2: the role of upper-case letters in propositional logic
(2) Logical Operators ( ~ ∙ ᴠ ⊃ ≡ )
The logical operators are perhaps the most important components of our formal language and
are used to represent the occurrence of important logical words in English. The occurrence,
placement, and relationship between these logical words and the simple, declarative, positive
sentences of English will determine an argument’s underlying logical form. The following chart
provides you with English’s important logical words and which of the five logical operators we
will use to represent them.
Logical
Word
not
Logical
Operator
~
(tilde)
Operator
Name
negation
Example
Translation
Moby Dick is not a whale
~W
and
∙
(dot)
conjunction
Socrates is mortal and
Aristotle is a philosopher
S∙A
or
ᴠ
(wedge)
disjunction
Socrates is mortal or
Aristotle is a philosopher
SᴠA
if … then …
⊃
(horseshoe)
implication,
conditional
If Moby Dick is a whale, then
Captain Ahab hates Moby
Dick
W⊃H
… if and
only if …
≡
(triple bar)
equivalence,
biconditional
Socrates is mortal if and only
if Aristotle is a philosopher
S≡A
Chart 3: the logical operators and their English counterparts
Let’s take this opportunity to introduce some terminology. We shall call upper-case letters by
themselves simple well-formed formulas (or WFFs) of propositional logic. And when we add
some logical operators to the mix, these are called complex WFFs. Now, there is an important
difference between ‘ ~ ’ and the other four logical operators: the ‘ ~ ’ operates on a single WFF
(monadic operator) while the other four logical operators connect two WFFs together (binary
operator). For this reason, these last four logical operators are called logical connectives.
(3) Parentheses ( … ) or [ … ]
The role of parentheses or brackets in propositional logic is to “group” parts of our WFFs together
in order to avoid ambiguity. As a general rule, you will need parentheses in your WFF if there are
more than two simple WFFs being used in your translation. Additionally, anytime you see English
words like ‘either’, ‘both’ or a comma, this is a strong indicator that you will need to “group”
parts of the WFF together. When and how parentheses are used is best conveyed by example
but the following chart offers some common English sentences that call for parentheses.
English
Either not A or B
Not either A or B
Both not A and B
Not both A and B
WFF
(~AᴠB)
~(AᴠB)
(~A∙B)
~(A∙B)
Chart 4: common English sentences that call for parentheses
Part 2: Formation Rules
Now that you have some appreciation of the role played by the propositional variables, logical
operators, and parentheses in propositional logic, we will need to outline the rules for “building”
complex WFFs out of simple WFFs. Just as there are rules for how to put parts of English
sentences together to form grammatically correct sentences, propositional logic also has rules to
the same end. Luckily for us, propositional logic is a much simpler language than English so its set
of formation rules are simple and straightforward (unlike the grammatical rules of English!).
▪
▪
▪
Rule (1): Any upper-case letter by itself is a WFF
Rule (2): Putting a ‘ ~ ’ in front of any WFF is a WFF
Rule (3): Connecting any two WFFs with a ‘ ∙ ’ ‘ ᴠ ’ ‘ ⊃ ’ or ‘ ≡ ’ is a WFF
We should understand Rule (2) pertaining to the ‘~’ as stating that this symbol can only ever be
placed in front of a WFF. In other words, it says that the ‘~’ can never be used to connect two
WFFs together. Additionally, we should understand Rule (3) as stating that the four other
mentioned symbols can only ever connect two WFFs together. None of these symbols are
allowed to “hang” off the end of a string of symbols or occur one right after another.
Part 3: Semantics in Propositional Logic
A semantics for propositional logic includes an assignment of a truth-function to each of our five
logical operators. Much like the functions you are familiar with from mathematics (e.g., addition,
subtraction, multiplication, division, etc.), a truth-function takes an input and produces a
corresponding output. Whereas in mathematics the input and output values are numbers, in logic
the input and output values are either true (T) or false (F). Below you will see each logical
operator’s truth table which is a complete specification of the truth-function assigned to the
symbol. Along with the definitions of validity and invalidity, the truth tables for the five logical
operators are absolutely essential to succeed in the logical study of argumentation.
With the symbolism, formation rules, and semantics of propositional logic in hand, you will soon
be able to use these tools to represent the underlying logical form of natural language arguments
(translation). With the logical form of an argument revealed, you will first learn how to employ
the truth tables or semantic tableaux to determine whether an argument is valid or invalid.